The long-term objective of this grant is to use the recent rapid advances in parallel computational hardware and associated software to enable new approaches to image science and image-quality assessment in the context of single-photon emission computed tomography (SPECT). To achieve this broad objective, research under this grant strives to develop new concepts, mathematical theories and models as well as the new computational hardware and algorithms needed to implement them. One major area of emphasis is on system models in which the object is described as a function of space and time rather than as a discrete array of voxels. In this approach, the system is described by an operator rather than a matrix, and we focus on exact mathematical descriptions of this operator either through analytical singular-value decomposition or by means of the Fourier crosstalk matrix. This analysis will allow determination of the inevitable null functions (invisible objects)for any particular system and will show how the null space can be controlled by the use of a positivity constraint in the reconstruction. The object function can be regarded as one sample function of a spatiotemporal random process, and another area of emphasis will be the full infinite-dimensional characterization of this random process through a concept called a characteristic functional. Knowledge of this functional will lead to an understanding of the statistics of features derived from a reconstructed image and to new methods for computing task-based metrics of image quality. The raw image data in SPECT and other photon-counting modalities is not a pixelized projection image but rather a list of attributes, such as position, energy and time of arrival, of each detected photon. The measured attributes can be regarded as samples from a complicated random point process, and we have derived the operator that relates the mean of this process to the spatiotemporal object function. We will study the properties of this operator in detail and relate it to image quality. This analysis will permit a rigorous study of the relation between radiation dose and image quality. Other studies will investigate new figures of merit for image quality and apply them to the optimization of SPECT systems and to the new field of adaptive SPECT. Computational power for these efforts will be obtained either through next-generation graphics processing units such as the Nvidia Maxwell or with arrays of CPUs and coprocessors. Parallel code to implement all of these advances will be made available through the web.